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Revista
Mexicana
de Física 39, No. 2 (J993)
314-328
On the radiation by a charge in a material medium
J.A.E.
ROA-NER1t.,
M. VILLAVICENCIO' AND J.L.
JIMÉNEZH
t Departamento
de Física, División de Ciencias Básicas e Ingeniería
Universidad Autónoma Metropolitana-Iztapalapa
Apartado postal 55-534, Og340 México D.F .
• Departamento de Física, Facultad de Ciencias
Universidad Nacional Autónoma de México
Apartado postal 21-939, 04000 México D.F.
Recibido el 14 de agosto de 1992; aceptado el 23 de octubre de 1992
In this work we obtain the exact non linear equation of motion, including radiation
reaction, for a charge moving in an isotropic, homogeneous and linear medium. From this equation
ABSTRACT.
we recover the known results for the dynamics of charges in vacuum and discuss the involved
approximations. Using the hypothesis of uniform motion in the exact non linear equation we
obtain the usua! results of the Cherenkov e!fect. Also we analyze severa! conceptual points in
arder to get a better understanding of the Cherenkov e!fect. \Ve study the energy balance and
show explicitly the existence of an external force to avoid a contradiction with the electromagnetic
theory; we show that, in this case, the charge behaves as a transducer transforming completely
mechanica! work into radiation. \Ve also discuss the physical origin of the Cherenkov radiation,
giving an explanation why there is radiation even when the charge is in uniform motion.
RESUMEN. Se obtiene la ecuación de movimiento no lineal, incluyendo reacción de radiación,
para una carga moviéndose en un medio lineal, homogéneo e isótropo. De esta ecuación se ob-
tienen los resultados conocidos para la dinámica de cargas en el vacío y se discuten las aproximaciones que se utilizaron para ello. Utilizando la hipótesis de movimiento uniforme en la ecuación no lineal se obtienen los resultados conocidos para el efecto Cherenkov. Se analizan
algunos aspectos conceptuales que ayudarán a un mejor entendimiento del efecto Cherenkov.
Se estudia el balance de energía y se muestra que en este caso debe existir una fuerza externa
para evitar una contradicción con la teoría electromagnética, es decir, en este caso la partícula
se comporta como un transductor que convierte completamente energía mecánica en radiación.
PACS: 03.50.De; 4LlO.-j; 41.70.+1
l. INTRODUCTION
One of the most important and well known results in classical clectrodynamics
is that,
in vacuum, a charge radiates only when it is accelerated. On the other hand, it is also a
well known result that when a charge travels with constant velocity through a material
medium, it radiates if its velocity is greater than the velocity of light in that medium;
that is v > cl Vi, where f is the dielectric function of the rnedium (we shall assume
.Ion leave from Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma
de México, Apartado postal 21-939,04000 México D.F.
ON THE RADlATION.
..
315
throughout this work J1. = 1, with J' the magnetic permeability). This phenomenon, known
as the Cherenkov effect, is of great importance in high energy physics where it is used
to determine the velocity of elementary particles. Owing to its usefulness this effect is
discussed in several texts of electromagnetism [1-71. In the early fifties Motz and Schiff [8]
studied briefly the Cherenkov effect in a dispersive medium; ten years later Volkoff [9)
clarified the apparent contradiction between the Cherenkov effect and Gauss's Law by
taking into account the discontinuous nature of the electric field on the Cherenkov cone.
In this excellent work it is evident the care needed to handle the electromagnetic fields
on the surface of the Cherenkov cone, in absence of dispersion, because of the extremely
singular behavior of the fields.
Sorne years ago two works appeared [10, 111 where the Cherenkov effect was discussed
completing the analysis avaHable in most common texts. Nevertheless, in the methods
presented in the literature there are several points which deserve a careful discussion. As
an example we can point out that in the method followed by Jackson [3], Panofsky and
Phillips [5], and Good and Nelson [61 it is not clear how radiation can be predicted for a
charge moving with constant velocity using Lienard- Wiechert potentials for an accelerated
charge. This is a common source of confusion for the student. Furthermore, there is not
an explicit mention of an external force needed to resto re the energy lost by radiation,
making difficult the understanding of the energy balance.
There are at least two methods to determine the energy radiated per unit time in the
Cherenkov effect. One is due to Landau [2), who equals the power radiated by the charge
with the work done in unit time by a stopping force eE self-exerted on the particle by the
self-field. The other method [4), nses Poynting's vector to determine the energy radiated
per unit time.
One of the most interesting and challenging problems that arises when the dynamics of a
charge is studied is the analysis of the effects produced by the radiated fields on its source,
that is, the radiation reaction. Hence it is interesting to discuss the dynamical effects on a
charge due to the emission of Cherenkov radiation. In the present work we make explicit
and thoroughly develop an implicit idea in the work of Landau [21 on the ionization
losses by fast particles in matter. This idea suggests us to face the Cherenkov effect as a
radiation reaction problem. This approach leads to a deep analysis of radiation reaction
beyond the usual treatments in the texts of electromagnetism [4, 5, 12-161 and journals
devoted to teaching. Then, we first discuss several physical aspects of the derivation of
the equation of motion for a radiating charge, and exhibit the usual approximations that
must be avoided to obtain a useful expression for the radiation reaction force for a charge
moving in a medium. Our approach shows clearly how a charge in uniform motion can
radiate and, as a particular case, we obtain the usual results of the Cherenkov effect.
Also we clarify the energy balance for a charge moving in a medium and sorne physical
aspects of the origin of Oherenkov radiation showing that if we assume a constant velocity
for the radiatillg particle, there must exist an external force acting on it, so in this case
we can think the particle as a transducer transforming completely mechanical work into
radiation.
It is interesting to remark that the results obtained with the hypothesis of constant
velocity in tbe discussion of the Cherenkov effect agree with usual experiments [17) beca use
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J.A.E.
ROA-NERI
ET AL.
the change in the velocity of the partic1e, traveling through a medium, is so small that
can be neglected for all practical purposes.
2.
THE
RADIATION
REACTION
IN A MATERIAL
MEDIUM
The starting point in the analysis of the dynamics of a charge in vacuum is the postulate
of the conservation of linear momentum of the system constituted by the charge, the fields
produced by it, and the external fields [41; this leads to the equation
Fex'
+ F""If
(1)
= moa.
Using the same reasoning it can be demonstrated that this equation is satisfied even when
the partic1e is moving through a dielectric medium. In this case the effects of the medium
are represented through the explicit form of each term in the preceding equation; for
example, the electromagnetic fields in F ex' and F ,elf are different from their respective
expressions in vacuum owing to the perturbation on the fields caused by the dielectrico
In order to determine the se1f-interaction force on a charge distribution p moving
through a dielectric we assume that it is spherically symmetric, non rotating, and that
its center follows the trajectory r(t), so the self-force can be expressed as
Fself(r, t) =
J
d 3xp(lx - r(t)I)
(
E,elf(X, t)
v(t)
+ -c-
x B,elf(X, t) ) ,
(2)
where E,e¡¡(x, t), B,elf(X, t) are the fields produced by the charge. It must be emphasized
that, unlike the self-fields in vacuum, these fields are not produced directly and exc1usively
by the charge; they are the inhomogeneous solutions to the macroscopic Maxwell equations
with the moving charge distribution as source. These fields contain then the response of
matter to the perturbation produced by the charge. In this sense it is c1early a self.
interaction phenomenon.
Since in the Cherenkov effect the velocity is assumed to be constant, we can follow two
possible ways to determine F ""If: we can use the frame in which v(t) # O, or we can use
the rest frame of the partic1e, where the medium moves with velocity -v(t), finding the
se1f-fields with the Minkowsky equations for moving mediums. For simplicity we follow
the first method.
In order to obtain the se1f.fields we work in the Coulomb gauge. In this gauge the fields
are determined by the transverse component of the vectorial potential A(x, t) by means
of the equations
18Al-
E""lf =
-~8t'
(3)
\Ve omit the contribution E' = -V' 4>e beca use, as is well known, in Coulomb gauge
4>e is an electrostatic potential and for a spherically symmetric charge distributions its
ON
THE
RADJATIONo
317
oo
contribution to the self-force vanishes [3, 4, 13]0At the same time the Fourier transform
of A.t, Á.dk, w), satisfies the equation
(4)
where
(5)
To obtain A(x, t) it is usually enough to take the inverse Fourier transform of Á(k,w);
this is possible since J(x, t), and consequently j(k, w), are known functionso Nevertheless,
when the radiation reaction problem is studied, as in this case, the trajectory of the
particle is unknown and then J(x,t) and j(k,w) are not specified beforehando Moreover,
it is risky to assume that the system of Maxwell's equations is complete even when J(x, t)
is unknown, but it is the usual and unique starting point with which we can count [18,
19lo It is evident that this fact deserves a careful and deep study, but now we use this
hypothesis in the same way as in the study of radiation reaction in vacuumo That is, we
consider J as unknown and at the end, in order to recover the usual results, we assume a
particular J = pv, with constant VoThe merits and deficiencies of this hypothesis will be
seen in the obtained resultso
As a point of departure we have the general solution to the Maxwell equations that
can be obtained from Eqo (4), that is
A(x, t) =Ahom
2
+ (27r)3/2
c
J' J J 3
dt
d k
dwp(k)
k kx
2
k)
(v(t') x
_ ~
o
(6)
e
o
ei[k.(x-r(t'»-w(t-t'»)
o
From this expression it is easy to get the fields which contain the contributions of the
charge and the medium,
2 2
E.e1f(X, t) = - (211')3/2
c
J'J
3
dt
d k
J -
dwp(k)
(-iWkX
k
(v(t')
2 _ ~
Xk))
o
(7)
e
o
ei[k.(x-r(t'))-w(t-t'))
,
and
2
B.elf(X, t) = (27r)3/2c
o
J' J J dt
ei[k.(x-r(t'»-w(t-,,»)
d 3k
dwp(k)
(ikk2 _x ~v(t')
e
o
)
o
(8)
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J.A.E.
ROA-NERI
ET AL.
The seU-force can be obtained directly from these equations. The electric and magnetic
contributions to it are, respectively,
F •• (r, t) = I{
!
C2
elee
¡¡ ¡
dt'
d3k
dwl¡3(kW
(-iWkk x (v~t') x k)).
2 _
w t:~w)
e
(9a)
. ei[k(r(t)-r(t'))-W(f-t')]
and
F ••If (r,
t)
=-
ma,
!2 ¡dt'
C
¡¡
d3k
dwl¡3(kWk
(iV(t)
x (k x
k2 _
V(t'))).
w2:~w)
(9b)
. ei[k.(r(t)-r(t'))-w(t-t')J.
The preceding expressions are as general as the solutions to the macroscopic Maxwell
equations and the Lorentz force. From them it is possible to get all the necessary information to discuss sorne special points of interest. Now, it is impossible to continue our
analysis because €(w) is unspecified, and the general characteristics of this function are
not enough to trace the most relevant aspects of the seU-force. But, using a particular
model for €(w), such as that of Drude-Lorentz, we can integrate it, in an approximated
way, over w. The results are useless for our purposes. For a thoroughly discussion of the
Cherenkov effect taking dispersion into account see Allison [17].
In order to reproduce the usual results for radiation reaction in vacuum [4, 5, 20-231,
and get a better idea of the approximations involved, it is enough to assume that €(w) is
constant. Although this supposition is unphysical, it permits us to integrate over w and
get sorne interesting results. After integrating we obtain
. eik.[r(f)-r(f')]
(10)
. eik.[r(t)-r(t')J,
where
e
is the Heaviside step function. It must be noted that in the limit
€ ~
1 this
equation does not reduced tú the usual equation ror a charge moving in vacuum [20-23],
beca use in that case the analysis of the radiation reaction is done in the rest frame of the
partic!e, that is v(t) = O,consequently there is not a magnetic contribution. Gn the other
hand, Eq. (8) is a non-linear one; this can be seen easily expanding the product
v( t')eik.[r( t)-r( t')]
ON THE RADlATION...
319
in a Taylor's series around time t. In this expansion terms appear involving products of
v(t) with its higher order derivatives or products of them, so to obtain a linear equation, as
in the usual analysis of radiation reaction in vacuum, it will be necessary to approximate
the exponential to unity. However, this approximation is not enough since in the magnetic
contribution the product v(t).v(t') appears. Then it is necessary to impose the restriction
v(t) = O, which means that the description is made in the rest frame of the charge. Both
approximations and the limit f ...•.•1 lead us to the usual results.
Carrying out all angular integrations in Eq. (10) and approximating the exponential to
unity lead to
F ••lr = - 3~;2
J
J
dt'a(t')
dkklPjt
sin cc(~
t'))e(t
- t'),
(11)
and defining
(12)
results in
2
32rr
F••lr = --3c
J ' ( ') (')e( ')
dt 9 t - t a t
t- t ,
(13)
which, in the limit f = 1, is exactly the linear integrodifferential equation for the radiation
reaction discussed in severa! papers [20-231. In the case of a point particle Eq. (13) reduces
to the famous Abraham- Lúrentz expression
(14)
where bm. is the e!ectromagnetic mass, which is 4/3 of the electrostatic energy divided
by c2, a result usually regarded as a contradiction with special relativity. The respective
expression in a non dispersive dielectric medium is
(15)
which does not give us more information that the equation in vacuum. In this case the
electromagnetic mass fbm. represents not only the contribution of the vacuum self-field
to the inertia, but also the contribution due to the medium.
3.
THE
CIIERENKOV
EFFECT
Although Eq. (13) is important because it led us to obtain a linear equation of motion
with the well known virtues of the equation of motion for extended charges in vacuum,
320
J.A.E.
ROA-NERI
ET AL.
it is only an approximated expression for the problem of self-interaction for a particle
moving through matter. If we want to obtain the usual results of the Cherenkov effect
from this equation we have to face a problem: it has been used the approximation v(t) = O
which hinders us to analyze the case of our interest, when v > c/../f. However, although
our non-linear Eq. (la), which is exact .and valid even for dispersive mediums, is unhandy,
it takes a simple form in the case of the Cherenkov effect where v is constant.
As we mentioned before, to arrive to Eq. (la) we assumed the consistency of Maxwell
equations even when the current density is unknown. Now, if we assume the particle moving with constant velocity its trajectory becomes known and we will be able to determine
F••Ir and the radiation emitted; as we shall see, with this result it is possible to understand
the energy balance.
Qne of the most famous derivations of the Abraham-Lorentz equation is that due to
Planck [241.This derivation assumes that the power radiated is equal to the work done
by the radiation reaction force per unit time, so if we use the known self-force express ion
with constant velocity in order to get the unknown power radiated, which in this case is
proportional to the radiated energy per unit length, we must obtain the usual expression
for the Cherenkov effect. As we shall see, this really happens. Although this idea is implicit
in Landau's work [2] it has not been developed. In this work we make it.
Taking into account the idea of Planck, our next step is to introduce in Eq. (la),
which contains the information of the dispersive medium and the non-linear terms, our
hypothesis of constant velocity for the particle. So
ei[k.(r(t)-r(t'))J
=
eik.v(t-t').
(16)
The integration over t' can be carried out and it leads us to b(w + k . v); without loss
of generality we can choose the particle moving in the x direction, then performing the
integrations we get
(l7a)
where
q2 = k2y +
e
,
and
F ••1f =
m.,
o.
(17b)
Because of the symmctry of the problem the transversal components to the \'elocity
are zero, and using d2k = q dq d<p and the dispersion relation w21k2 = c2/«w), Eq. (17a)
ON THE RADIATION...
321
transforms into
82'
F ••11 =
1
1r 2 V
J J
dw
ve
elec
1
w(
2
dq ql¡i(k)1
c')
-V"'<\wj
(
)
w'
q+",-I-c
2
(18)
•
t12f~W)
In the limit of a point particle it reduces to
ie2 y
F:;;; = 1re2;
J J
dw
c' )
1 _
(
dqqw
2
V"'<\wj
(19)
V"\W») .
w' (
q+",-I-c
This expression coincides with that obtained by Landau [2].
It must be emphasized that if a non dispersive medium is considered [«w) is constant],
the integrand of the preceding equations depends in a crucial way on the magnitud e of
the velocity of the particle with respect to velocity of light. If the speed of the particle is
less than c/";<, the integrand is an odd function with no poJes, and the integration limits
are symmetric so F ••Ir = O;on the other hand, if the speed of the particle is greater than
el";<, the integrand have poles and the integral ean be different from zero. \Ve have then
as a consequence that if the particle travels with constant velocity in vacuum or through
a medium with velocity v < el";< it does not radiate, but when its velocity is v > el";<
in the medium, or if it eould travel in vacuum faster than light, then there would be
radiation producing a self-force different from zero, F ••Ir f O.
In the dispersive case the self-force depends on the detailed behaviour of «w), so without
a model for it we can not obtain general conclusions. In fact we ean see from Eq. (19) that
the specilic behaviour of v2«w)/c2 determine if there are or not poJes in that equation.
If the velocity of the charge is constant it implies that the power radiated is
dE
P =
dE
= v dx = -F ••Ir'
di
(20)
Y,
then
2
dE
ie
dx = - 1rC2
J (1dww
2
c)
v2«w)
J
dq
2
+ w'
q;T
(1q _ V,,\W»)
.
(21 )
e
Performing the integration over q we get
2
dE
e
dx = c2
J (
dww
2
c
)
1 - v2«w)
,
which is precisely the usual result [3, 251 for the Cherenkov effect.
(22)
322
J.A.E.
ROA-NERI ET AL.
On the other hand, when we integrate Eq. (10) over t' we get a Dirac's delta function
which impose the condition w2 = (k. v)2 = k2v2 cosll. This condition and the dispersion
relation imply that
(23)
or
cos IIc =:!:
e
r,'
vv(
(24)
So we have obtained the Cherenkov angle, 1Ic,which defines the cone within which there
can be radiation. \Ve must remark that this result is independent of the charge distribution. However, the spectral distribution depends on the charge distribution [25].
Another point that deserves special attention is the following: Planck's idea about
radiation reaction, according to which the power radiated is equal to the work done by
the self force per unit time, plays a fundamental role in the determination of the energy
lost per unit length in Cherenkov radiation. As it has been shown in another paper [26] this
idea, which seems so reasonable and suggestive, is in general wrong in vacuum. However,
in this case it has given excellent results; it is so, as we shall see, due to the presence of
a medium and the uniform motion of the charge.
As we have seen, the existence of radiation when the velocity of the particle is greater
than el Vi has as a consequence the existence of a radiation reaction force that disappears
when v < el Vi. In this way it is clear that if the Cherenkov effect ean be faced as a
radiation reaction prablem, it is also clear that the relation with the radiation reaction
in vacuum is not immediate; the most relevant differences reside in the linear character
of the usual equation of motion in vacuum. Besides, in vacuum the trajectory and the
velocity of the particle are unknown even when the external force is known, while in the
Cherenkov effect the velocity is known. This is precisely the point that we shall discuss in
the following section fram the perspective of the energy balance. As we see, the Cherenkov
effect considered as a radiation reaction problem leads to the problem of radiation reaction
itself, a yet properly unsolved classical problem.
4. ENERGY BALANCE
As we can see from the radiation reaction problem in material mediums, even when it is
assumed a constant velocity for the particle there is radiation if the speed of the particle
is greater than el Vi. However, it is also important to remark that without the presence
of an external force acting on the particle there will be an obscure source of energy that
restores the radiated energy. This fact is the basic idea in Planck's theory of the radiation
reaction. \Ve can show easily that in the Cherenkov elfect it is strictly neeessary the
presenee of an external force to resto re the energy lost by the charge. To see this we will
use Poynting's express ion for the energy balance [3],
dUmec _
----dt
Jd3 x.J E -
fs
dUelm
.dS ur+-dt
'
(25)
ON THE RADIATION...
323
where, as usual, the first ter m on the right side is related to the power radiated,
p=
f
S.dSur.
(26)
Let us first analyze the left hand side of Eq. (25). It is elear that J is the current
density, which in this case is the convected current produced by the charge itself in
uniform motion, that is J = pv. E is the electric field and it is necessary to elarify what
kind of field we are working with. In sorne modern texts [41 where Poynting's theorem is
derived, E is an external electric field, that is a field not produced by J itself. However, this
point of view has sorne problems [27]. In other deductions Eq. (25) is obtained through a
formal manipulation of the Maxwell equations, so E represents the general solution to the
inhomogeneous Maxwell equations and then ineludes not only the homogeneous solution
Eh, that can be associated with external fields, but the particular solution Ei = E.elf,
which is associated with the fields produced by J. So we have that
(27)
but
as
we have assumed v = constant
dUmec = O
dt
and then
Fext = -Fself
where the external force can have in general an electromagnetic and a non electromagnetic
contribution. Using this result in Eq. (25) gives us
p = _ dUelm
dt '
(28)
and from Eq. (20)
-F.elf'
v
= Fext . v = -~dUelm = P.
(29)
It is interesting to remark that although a charge in uniform motion in a medium
produces static fields, these fields induce a time dependent polarization in the medium,
due to the change with time of the distance from the charge to every element of the
medium. So Uelm varies with time.
Now we have three interesting points: In the first place we can see how Jackson's
approach to the Cherenkov effect is also a radiation damping problem, hence the full
equivalence of Landau's and Jackson's approaches, as we could expect; in the second
place, from Eq. (29) we can conelude that if P f O then there must necessarily exist an
324
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ROA-NERI
ET AL.
external force which restores to the charge the energy lost by radiation. This external
force can be mechanical or electromagnetic,
but if it does not exist it is impossible to
maintaiD a CODstaDt velocity for the particle; fiDally, we have ShOWDEq. (29) to be a
consequeDce of POYDtiDg'S theorem, so in this case we show PlaDck's conjecture is correct
and the charge behaves as a traDsducer which transform completely the energy obtaiDed
from the external force iDtO radiatioD. This situatioD does DOt appear iD vacuum, being
Schott's term a prove of it. This term is iD part respoDsible for the confusions with the
energy balance in the radiatioD reaction problem in vacuum. \Ve must emphasize that
this cODspicuous fact is not discussed iD most texts on electromagnetism
[4, 5, 12-16,
281. It must be also remarked that our Eq. (10) remains valid eveD wheD the velocity of
the charge chaDges; iD real experimeDts [171 there is not an external force to restore the
energy lost by radiatioD, but the chaDge in the velocity of the charge is so small that the
hypothesis of constaDt velocity is quite well.
5. VELOCITY
AND RADlATION FIELDS
Although it is apparent the traDsformatioD of mechanical eDergy iDtO radiation even with
a CODstaDt velocity of the charge, it is DOt clear the relation between Cherenkov radiatioD
and the usual assumptioDs used to analyze the motion of a charge in vacuum. \Vhen the
radiation emitted by a charge moving in vacuum is studied it is strongly remarked that
the radiatioD field varies as Ix - r'l-1 and depends on the acceleration, meanwhile the
velocity field varies as Ix - r'1-2 and it is considered as a field bounded to the particle.
The root of many of the difliculties in the understanding of the Cherenkov effect lies precisely in this separatioD between the velocity and radiation fields, since in the Cherenkov
effect the radiated field depends on the velocity. According to this fact, it is necessary to
clarify this appareDt contradiction.
ID order to clarify the problem it will be useful to analyze the behaviour of the electromagnetic potential in terms of the ratio of the velocity of the charge to the velocity
of light in the dielectric. In order to do it, it will be enough to consider a non dispersive
medium. ID the space (x, t) the electromagnetic potential Al.(x, t) satisfies the equation
(30)
for a point particle traveling with constant velocity the current density J 1. is proportional
to 6(x - vt). If, as we did before, we take the motion in the x direction, then
(31 )
we can expect the same functional
dependence
for the potential
Al.(x, t) = Al.(x - vt, y, z,).
Al., namely,
(32)
ON THE RADIATION..
•
325
This means that at time t + to the field has the same functional form that it had in the
previous position -vto, so the potential must satisfy
(33)
This relation permit us to write the expression (30) in the form
(34)
where
As we can see immediately, the electromagnetic potential A behaves in a completely
different way if the particle's velocity is less or greater than el Vi, since if the coefficient
of the second time-derivative is positive the preceding differential equation is an elliptical
one, of Poisson 's type. This fact is characteristic of the three dimensional electrostatic
behaviour. It is convenient to define the variable
(35)
in terms of which the Eq. (34) can be written as
(36)
On the other hand, if v > el Vi, the equation will be an hyperbolic one, of D'Alembert's
type, but in two spatial and one time dimensions. In this way the behaviour of the wave
motion depends notably on the number of spatial dimensions; indeed, we have seen that
in the Cherenkov effect the propagation of radiation is quite different from that in three
spatial dimensions.
By means of the analysis of the wave equation, in two or three dimensions [6), we
can give a better idea of the preceding discussion. The Cauchy solution of the three
dimensional wave equation shows that a perturbation produced at time t' and at position
r' travels in space like an spherical wave front centered in r', without changing its form,
but with its size diminished in such way that behind the wave front there is not any
perturbation at aH. Of course the wave front emerges from r'. In the bidimensional case,
if a perturbation is produced it will travel without changing its form, but there will
exist a remanent wake behind the wave front. This kind of behavior is also found in the
inhomogeneous solution of the wave equation. These differences in the wave behaviour
can be easily appreciated from their respective Green functions: in the three dimensional
case the Green function is Ix - r'1-16(t' + !Ix - r'l/c) - t), which shows the change in
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ET AL.
the amplitude and how the pulse is concentrated in a sphere, defined by the argument of
the Ó function, that grows up with time; furthermore the perturbation
at (x, t) depends
only on what happened in just one point in the past (r', t'). In the bidimensional case the
Green function is 2c[c2(t-t')2-lx-r'12j-l/2.8[e(t-t')-lx-r'lJ,
which, like in the three
dimensional case, shows a singularity in Ix - r'l = e(t - t'), but for Ix - r'l < e(t - t') there
is a wake, a remanent perturbation.
In the case of Cherenkov radiation the field at the
time t and position x is a consequence of what happened in the interior regio n defined by
the Heaviside funetion, whose boundary defines an edge called the Cherenkov eone that
at the same time represents indeed the radiation.
According to the preceding discussion, when the velocity of the eharge is v < e/0
we have the usual electrostatie behaviour that varies as Ix - r'I-2, but when v > e/0
the behaviour is ruled by a bidimensional wave equation whose dependence on Ix - r'l
ehanges drastieally.
The Green function reflects the physical fact that all the elements of the medium inside
the cone contribute to the radiation: the envelope of all the wave fronts produced by every
radiating dipole forms the Cherenkov radiation front. In vacuum, the Green function in
three dimensions shows how the charge in motion is the unique so urce of radiation.
As we have shown, in the Cherenkov effect there are not contradictions.
If v < e/0
we have an usual static behaviour, but if v > c/0 the behaviour of the electromagnetic
potential is ruled by the bidimensional wave equation whose characteristics are remarkably
different from those of the three dimensional wave behaviour. From the mathematical
point of view, the change in the field behaviour when the particle goes from the case
of "no propagation"
(no radiation) to the case of "propagation"
(radiation), previously
related to the presence or lack of poles in Eq. (19), now appears in the electromagnetic
potentials as a change in the behaviour of the functions describing it. In the first case they
are described in terms of modified Hankel functions K, with an asymptotic behaviour that
falls off rapidly with Ix - r'l - 1, while in the second case the potentials are described
in terms of Hankel funetions H which asymptotically vary as Ix - r'I-1/2, but since we
are working with a bidimensional problem, then in this case we have a typical radiative
behaviour.
6. CONCLUSIONS
The Cherenkov effect permits us to analyze several physical aspects of classical electrodynamics, in particular the challenging problem that arises from considering the effect
produced on the dynamics of a charge when we take into account the radiation produced
by it. It must be emphasized that the usual approaches to radiation reaction are inadequate to face the Cherenkov effect since the common linear approximation
is too coarse
to make the jobo Our analysis shows several interesting and coneeptually difficult points,
like those in the derivation of the radiation damping force itself, and the remarkable
eompatibility
of the energy balance for a radiating eharge in a medium and Planek's
approach in vacuum. Also our approach makes apparent that the Cherenkov effect is a
good example where the distinction between self-field and externa! fields in Poynting's
theorem is significant [27], and clarifies the energy balance in the Cherenkov effect. Hence
ON THE RADIATlON...
327
this effect becomes a problem of great conceptual and didactic importance, as we have
seen throughout this work. We have elarified in an elementary way most of the difliculties
the students face in studying the Cherenkov effect: the apparent contradiction
between
Cherenkov radiation and the usual radiation criterio n used in the analysis of radiation in
vacuum, within the framework of elassical electrodynamics.
On the other hand, the energy balance used in electromagnetism
has some difliculties,
mainly when we inelude the radiation reaction force in the analysis of the motion of
charges.
ACKNOWLEDGEMENTS
We want to express our gratitud e to 1. Campos and D. Moreno for their careful reading of
the manuscript and helpful discussions. J.A.E.R. acknowledges the support of DGAPAUNAM and the partial support of CONACYT, during the first part of this work. M.V.
acknowledges the support of DGAPA-UNAM.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
lO.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
A. Sommerfeld, Opties; Lectures on Theoretieal Physies, Vol. IV, Academie Press, New York
(1953) 328-336.
L.D. Landau and E.M. Lifshitz, Eleetrodynamies
o/ Continuo,," Media, Adisson- Wesley,
Reading Massachusetts (1960) 345, 350, 357-359.
J.D. Jackson, Classieal Electrodynamies, Wiley, New York (1962) pp. 494-499, 783.
J.D. Jackson, Classieal Electrodynamies, 2nd. ed., Wiley, New York (1975) 628-641, 783, 789.
W. Panofsky and M. Phillips, Classieal Eleetrieity and Magnetism, 2nd. ed., Adisson-Wesley,
Reading Massachusetts (1962) pp. 373-375, 387-390.
R.H. Good and T.J. Nelson, Classieal Theory o/ Eleetrie and Magnetie Fie/ds, Academic, New
York (1971) pp. 314-316, 534-536.
A.M. Portis, Eleetromagnetie Fie/ds Sources and Media, Wiley, New York (1978) pp. 567-576.
H. Motz and L.!. Schiff, Am. J. Phys. 21 (1953) 258.
G.M. Volkoff, Am. J. Phys. 31 (1963) 601.
H.A. Haus, Am. J. Phys. 54 (1986) 1126.
M. Bornatid and M. Spada, Am. J. Phys. 57 (1989) 634.
F. Rohrlich, Classieal Charged Partid •• , Adisson.Wesley, Reading Massachusetts (1965) pp.
15, 114-120.
C.P. Enz (editor), Eleetrodynamies.
Pauli Leetures on Physies, Vol. 1, MIT, Cambridge
Massachusetts (1973) p. 146-152.
L. Eyges, The Classieal Electromagnetie Fie/d, Dover, New York (1980) pp. 297-300.
J.B. Marion, Classieal Electromagnetie Radiation, Academic, New York (1980) pp. 305-330.
E.J. Konopinski, Electromagnetie Fie/ds and Re/ativistie Partid •• , McGraw-HilI, New York
(1981) pp. 441-442.
W.W.M. Allison and J.H. Cobb, Ann. Rev. Nud. Parto Sei. 30 (1980) 253.
M. Bunge, Foundations o/ Physies, Springer, Berlin (1967) pp. 164, 174-176.
J.L. Jiménez, Ciencia 40 (1989) 257.
D. Bohm and M. Weinstein, Phys. Rev. 74 (1948) 1789.
H.M. Fran~a, G.C. Marques and A. J. da Silva, Nuovo Cimento A 48 (1978) 65.
L. de la Peña, J.L. Jiménez and R. Montemayor, Nuovo Cimento B 69 (1982) 71.
328
J.A.E.
ROA-NERI
ET AL.
23. J.L. Jiménez and R. Montemayor, Nuovo Cimento B 73 (1983) 246.
24. M. Planck, Berl¡ner Berichte (1896) pp. 151-170; Ann. d. Phys. 60 (1897) 577.
25. M. Villavicencio, El Efecto Cherenkov como un Problema de Reacción de Radiación, Tesis,
Departamento de Física, Facultad de Ciencias. U.N.A.M. (1988) (unpublished).
26. 1. Campos and J.L. Jiménez, Am. J. Phys. 57 (1989) 610; Am. J. Phys. 55 (1987) 1017.
27. 1. Campos and J.L. Jiménez, Eur. J. Phys. 13 (1992) 117.
28. V.V. Barger and M.G. Ols80n, Classical Electricity and Magnetism. A Contemporar¡¡ Perspective, Allyn and Bacon, Reading Massachusetts (1987) pp. 460-462.